The Importance of Irrationality

The video tutorial explores how irrational ratios of frequencies in repeating patterns can create quasi-periodic arrangements, leading to new atomic symmetries previously thought impossible. It highlights the importance of irrational numbers, particularly the golden ratio, in achieving these symmetries, contrasting them with periodic patterns that rely on rational ratios. The tutorial emphasizes the role of quasi-periodicity in enabling fivefold and icosahedral symmetries, which were once considered forbidden by traditional crystallography.

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